Driving Home Faster After Dinner at Mom and Dad's

Suppose it took you 30 minutes less to drive the 40 miles back home after dinner at your parents' house than it did for you to arrive there in rush-hour traffic. You drove about 15 mph faster on the return trip. How can we analyze this scenario and what can we learn about the relationship between driving time, speed, and distance?

Let's break this down step-by-step:

The Scenario

• Distance to parents' house: 40 miles
• Time to arrive at parents' house during rush hour: Unknown
• Time to return home after dinner: 30 minutes less than the trip to the parents' house
• Driving speed on the return trip: 15 mph faster than the trip to the parents' house

Calculating the Driving Times

We know that the return trip took 30 minutes less than the trip to the parents' house. This means that if the trip to the parents' house took t minutes, the return trip took t - 30 minutes.

We also know that the driving speed on the return trip was 15 mph faster. Let's call the speed on the way to the parents' house v mph, and the speed on the return trip v + 15 mph.

The distance traveled is the same both ways (40 miles), so we can set up an equation:

• Time to parents' house: 40 miles / v mph = t minutes
• Time to return home: 40 miles / (v + 15) mph = t - 30 minutes

Solving for t, we get:

• t = 40 / v
• t - 30 = 40 / (v + 15)

Equating the two expressions for t, we get:

• 40 / v = 40 / (v + 15) + 30

Solving this equation, we find that v = 45 mph. This means that the driving speed on the way to the parents' house was 45 mph, and the driving speed on the return trip was 60 mph (45 mph + 15 mph).

Calculating the Driving Times

Now that we know the driving speeds, we can calculate the actual driving times:

• Time to parents' house: 40 miles / 45 mph = 0.889 hours = 53.3 minutes
• Time to return home: 40 miles / 60 mph = 0.667 hours = 40 minutes

The difference between the two trip times is 53.3 minutes - 40 minutes = 13.3 minutes, which is close to the 30 minutes we were told about in the original scenario.

Conclusion

This analysis shows how we can use the given information about the difference in driving time and speed to calculate the actual driving speeds and times for the two trips. By setting up equations and solving for the unknown variables, we can gain a better understanding of the relationship between driving time, speed, and distance, even in a scenario with some incomplete information.